One-sided Davis inequality for (F4) filtrations

TL;DR

This paper extends the classical Davis inequality to (F4) doubly indexed filtrations, proving a new inequality under minimal assumptions, which advances understanding of martingale inequalities in complex filtration structures.

Contribution

The paper proves a Davis-type inequality for (F4) filtrations without restrictive assumptions, broadening the scope of martingale inequalities.

Findings

Established the inequality under (F4) condition.

Removed the need for regularity or strong martingale assumptions.

Extended classical martingale inequalities to complex filtration structures.

Abstract

The classical Davis inequality $\mathbb{E} Mf\simeq \mathbb{E} Sf$, where $(Sf)^2=\sum_{k}\left|f_{k}-f_{k-1}\right|^2$ is the square function and $Mf= \sup_n \left|f_n\right|$ is the maximal function, is true with a universal constant for any martingale $f$ on any filtration. A natural analog in the setting of (F4) doubly indexed filtrations, i.e. $\left(\mathcal{F}_{i,j}\right)_{i,j}$ such that the operators $\mathbb{E}\left(\cdot\mid \mathcal{F}_{i,\infty}\right)$ and $\mathbb{E}\left(\cdot\mid \mathcal{F}_{\infty,j}\right)$ commute and their product is $\mathbb{E}\left(\cdot\mid \mathcal{F}_{i,j}\right)$, is the conjecture \[\mathbb{E}\sup_{n,m} \left|f_{n,m}\right|\simeq\mathbb{E}\left(\sum_{i,j}\left|\Delta f_{i,j}\right|^2\right)^\frac{1}{2},\] where $\Delta f_{i,j}=f_{i,j}-f_{i-1,j}-f_{i,j-1}+f_{i-1,j-1}$. It was known to be true only with some highly restrictive additional assumptions, e.g. regularity of the filtration ($g_{n,m}\gtrsim g_{n+1,m},g_{n,m+1}$ for any positive martingale $g$) or $f$ being a strong martingale ($\mathbb{E}\left(\Delta f_{i,j}\mid \mathcal{F}_{i-1,j}\vee \mathcal{F}_{i,j-1}\right)=0$). We prove the inequality $\lesssim $ assuming just the (F4) condition.